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DOI: 10.4230/LIPIcs.SoCG.2023.47

Sparse Euclidean Spanners with Optimal Diameter: A General and Robust Lower Bound via a Concave Inverse-Ackermann Function

Authors: Hung Le, Lazar Milenković, and Shay Solomon

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract

In STOC'95 [S. Arya et al., 1995] Arya et al. showed that any set of n points in ℝ^d admits a (1+ε)-spanner with hop-diameter at most 2 (respectively, 3) and O(n log n) edges (resp., O(n log log n) edges). They also gave a general upper bound tradeoff of hop-diameter k with O(n α_k(n)) edges, for any k ≥ 2. The function α_k is the inverse of a certain Ackermann-style function, where α₀(n) = ⌈n/2⌉, α₁(n) = ⌈√n⌉, α₂(n) = ⌈log n⌉, α₃(n) = ⌈log log n⌉, α₄(n) = log^* n, α₅(n) = ⌊ 1/2 log^*n ⌋, …. Roughly speaking, for k ≥ 2 the function α_{k} is close to ⌊(k-2)/2⌋-iterated log-star function, i.e., log with ⌊(k-2)/2⌋ stars. Despite a large body of work on spanners of bounded hop-diameter, the fundamental question of whether this tradeoff between size and hop-diameter of Euclidean (1+ε)-spanners is optimal has remained open, even in one-dimensional spaces. Three lower bound tradeoffs are known: - An optimal k versus Ω(n α_k(n)) by Alon and Schieber [N. Alon and B. Schieber, 1987], but it applies to stretch 1 (not 1+ε). - A suboptimal k versus Ω(nα_{2k+6}(n)) by Chan and Gupta [H. T.-H. Chan and A. Gupta, 2006]. - A suboptimal k versus Ω(n/(2^{6⌊k/2⌋}) α_k(n)) by Le et al. [Le et al., 2022]. This paper establishes the optimal k versus Ω(n α_k(n)) lower bound tradeoff for stretch 1+ε, for any ε > 0, and for any k. An important conceptual contribution of this work is in achieving optimality by shaving off an extremely slowly growing term, namely 2^{6⌊k/2⌋} for k ≤ O(α(n)); such a fine-grained optimization (that achieves optimality) is very rare in the literature. To shave off the 2^{6⌊k/2⌋} term from the previous bound of Le et al., our argument has to drill much deeper. In particular, we propose a new way of analyzing recurrences that involve inverse-Ackermann style functions, and our key technical contribution is in presenting the first explicit construction of concave versions of these functions. An important advantage of our approach over previous ones is its robustness: While all previous lower bounds are applicable only to restricted 1-dimensional point sets, ours applies even to random point sets in constant-dimensional spaces.

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Hung Le, Lazar Milenković, and Shay Solomon. Sparse Euclidean Spanners with Optimal Diameter: A General and Robust Lower Bound via a Concave Inverse-Ackermann Function. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 47:1-47:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{le_et_al:LIPIcs.SoCG.2023.47,
  author =	{Le, Hung and Milenkovi\'{c}, Lazar and Solomon, Shay},
  title =	{{Sparse Euclidean Spanners with Optimal Diameter: A General and Robust Lower Bound via a Concave Inverse-Ackermann Function}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{47:1--47:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.47},
  URN =		{urn:nbn:de:0030-drops-178976},
  doi =		{10.4230/LIPIcs.SoCG.2023.47},
  annote =	{Keywords: Euclidean spanners, Ackermann functions, convex functions, hop-diameter}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2022.54

Sparse Euclidean Spanners with Tiny Diameter: A Tight Lower Bound

Authors: Hung Le, Lazar Milenković, and Shay Solomon

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract

In STOC'95 [ADMSS95] Arya et al. showed that any set of n points in R^d admits a (1+ε)-spanner with hop-diameter at most 2 (respectively, 3) and O(n log n) edges (resp., O(n log log n) edges). They also gave a general upper bound tradeoff of hop-diameter at most k and O(n α_k(n)) edges, for any k≥2. The function α_k is the inverse of a certain Ackermann-style function at the ⌊k/2⌋th level of the primitive recursive hierarchy, where α₀(n)=⌈n/2⌉, α₁(n)=⌈√n⌉, α₂(n)=⌈log n⌉, α₃(n)=⌈log log n⌉, α₄(n)=log^* n, α₅(n)=⌊1/2 log^*n⌋, .... Roughly speaking, for k≥2 the function α_{k} is close to ⌊(k-2)/2⌋-iterated log-star function, i.e., log with ⌊(k-2)/2⌋ stars. Also, α_{2α(n)+4}(n)≤4, where α(n) is the one-parameter inverse Ackermann function, which is an extremely slowly growing function. Whether or not this tradeoff is tight has remained open, even for the cases k=2 and k=3. Two lower bounds are known: The first applies only to spanners with stretch 1 and the second is sub-optimal and applies only to sufficiently large (constant) values of k. In this paper we prove a tight lower bound for any constant k: For any fixed ε>0, any (1+ε)-spanner for the uniform line metric with hop-diameter at most k must have at least Ω(n α_k(n)) edges.

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Hung Le, Lazar Milenković, and Shay Solomon. Sparse Euclidean Spanners with Tiny Diameter: A Tight Lower Bound. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 54:1-54:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{le_et_al:LIPIcs.SoCG.2022.54,
  author =	{Le, Hung and Milenkovi\'{c}, Lazar and Solomon, Shay},
  title =	{{Sparse Euclidean Spanners with Tiny Diameter: A Tight Lower Bound}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{54:1--54:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.54},
  URN =		{urn:nbn:de:0030-drops-160629},
  doi =		{10.4230/LIPIcs.SoCG.2022.54},
  annote =	{Keywords: Euclidean spanners, hop-diameter, inverse-Ackermann, lower bounds, sparse spanners}
}

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DOI: 10.4230/LIPIcs.ITCS.2022.96

Dynamic Matching Algorithms Under Vertex Updates

Authors: Hung Le, Lazar Milenković, Shay Solomon, and Virginia Vassilevska Williams

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract

Dynamic graph matching algorithms have been extensively studied, but mostly under edge updates. This paper concerns dynamic matching algorithms under vertex updates, where in each update step a single vertex is either inserted or deleted along with its incident edges. A basic setting arising in online algorithms and studied by Bosek et al. [FOCS'14] and Bernstein et al. [SODA'18] is that of dynamic approximate maximum cardinality matching (MCM) in bipartite graphs in which one side is fixed and vertices on the other side either arrive or depart via vertex updates. In the BASIC-incremental setting, vertices only arrive, while in the BASIC-decremental setting vertices only depart. When vertices can both arrive and depart, we have the BASIC-dynamic setting. In this paper we also consider the setting in which both sides of the bipartite graph are dynamic. We call this the MEDIUM-dynamic setting, and MEDIUM-decremental is the restriction when vertices can only depart. The GENERAL-dynamic setting is when the graph is not necessarily bipartite and the vertices can both depart and arrive. Denote by K the total number of edges inserted and deleted to and from the graph throughout the entire update sequence. A well-studied measure, the recourse of a dynamic matching algorithm is the number of changes made to the matching per step. We largely focus on Maximal Matching (MM) which is a 2-approximation to the MCM. Our main results are as follows. - In the BASIC-dynamic setting, there is a straightforward algorithm for maintaining a MM, with a total runtime of O(K) and constant worst-case recourse. In fact, this algorithm never removes an edge from the matching; we refer to such an algorithm as irrevocable. - For the MEDIUM-dynamic setting we give a strong conditional lower bound that even holds in the MEDIUM-decremental setting: if for any fixed η > 0, there is an irrevocable decremental MM algorithm with a total runtime of O(K ⋅ n^{1-η}), this would refute the OMv conjecture; a similar (but weaker) hardness result can be achieved via a reduction from the Triangle Detection conjecture. - Next, we consider the GENERAL-dynamic setting, and design an MM algorithm with a total runtime of O(K) and constant worst-case recourse. We achieve this result via a 1-revocable algorithm, which may remove just one edge per update step. As argued above, an irrevocable algorithm with such a runtime is not likely to exist. - Finally, back to the BASIC-dynamic setting, we present an algorithm with a total runtime of O(K), which provides an (e/(e-1))-approximation to the MCM. To this end, we build on the classic "ranking" online algorithm by Karp et al. [STOC'90]. Beyond the results, our work draws connections between the areas of dynamic graph algorithms and online algorithms, and it proposes several open questions that seem to be overlooked thus far.

Cite as

Hung Le, Lazar Milenković, Shay Solomon, and Virginia Vassilevska Williams. Dynamic Matching Algorithms Under Vertex Updates. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 96:1-96:24, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{le_et_al:LIPIcs.ITCS.2022.96,
  author =	{Le, Hung and Milenkovi\'{c}, Lazar and Solomon, Shay and Vassilevska Williams, Virginia},
  title =	{{Dynamic Matching Algorithms Under Vertex Updates}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{96:1--96:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.96},
  URN =		{urn:nbn:de:0030-drops-156923},
  doi =		{10.4230/LIPIcs.ITCS.2022.96},
  annote =	{Keywords: maximal matching, approximate matching, dynamic algorithm, vertex updates}
}

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Sparse Euclidean Spanners with Optimal Diameter: A General and Robust Lower Bound via a Concave Inverse-Ackermann Function

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Sparse Euclidean Spanners with Tiny Diameter: A Tight Lower Bound

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Dynamic Matching Algorithms Under Vertex Updates

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